Dedicated IP's are desirable if you wish to access your website directly (without DNS resolution): "http://IP", and/or if you prefer a separate stream of DNS activity to your website. Accessing your website in this fashion is not possible with a shared IP, as multiple websites are accessed through it.
Once your domain resolves to our servers, in general there is no discernible difference between dedicated IP and non-dedicated IP accounts - with some exceptions. One such exception would be for email hosting - a shared IP means shared traffic via DNS. So, if one of the hosted email accounts on a shared IP sends an excessive number of emails, it may impact the availability or performance of the other accounts. For this reason, the expected activity of shared IP accounts is established through understanding the business needs of a given client, and also by bandwidth monitoring if it is not immediately obvious. We contact clients to register a dedicated IP address if their shared IP activity has potential to negatively impact other clients.
Previously, a dedicated IP was required to have an SSL certificate for your site. The rise and adoption of Server Name Indication (SNI) technology resolved this.